How do I Fill a Subdivided / Smoothed Barrel's cutout With 2 Sided Right Angle Face?
I'd do this in several stages, to make sure the topology stays nice and works well with both the subdivision modifier and the smooth shading.

Start with filling the horizontal surface. Connect the two verts that delimit this part and join them into an edge with F.
Subdivide this edge 5 times with W --> Subdivide --> crank up to 5 in the tools panel.
Then fill this surface with a grid fill (Spacebar menu --> Grid fill), using 6 spans and an offset of 6.


Repeat this process for the upper cap (6 spans and an offset of 3 for the grid fill this time):


Add a loopcut in the perimeter of the barrel, at the same height as the edge we started with in the last step. Create faces for the rim at the left and right by selecting 4 verts and filling a face (F).



Now you can fill the last hole with another grid (see span and offset params in gif below). Add another cut by selecting the two verts near the bottom and creating and edge with join J, to maintain a nice flow of vertical loops. Select the loop above which is distorted due to the current topology, and straighten it this way: make the leftmost vertex, that representing the desired height for all the verts, active by deslecting and reselecting it (Shift + RMB). Change the pivot point to active element, then scale all verts to 0 on the Z axis (S --> Z --> 0).



Now, if you like, you can add more loopcuts or increase the crease value to control the curvature of the subdivided surface.